how do i solve this equation with out using the quadratic formula:
5x^2 +18x +11 = 0
With those numbers, I don't see why one would not want to use the quadratic formula, but...
Guess one could use completing the square:
5x^2 +18x +11 = 0
x^2 + (18/5)x = -11/5
x^2 + (18/5)x + 81/25 = -11/5 + 81/25
(x + 9/5) = 26/25
x = (-9 ± √26)/5
To solve the quadratic equation 5x^2 + 18x + 11 = 0 without using the quadratic formula, we can try factoring or completing the square.
First, let's see if we can factor the equation. For a quadratic equation to be factorable, the numbers that multiply to give the constant term (11 in this case) should also add up to give the coefficient of the middle term (18 in this case).
Let's look at all the possible factor pairs for 11:
1 * 11
(-1) * (-11)
None of these pairs will add up to 18, so factoring is not a straightforward option.
Next, we can try completing the square. This method involves converting the equation into a perfect square trinomial and then solving for x. The general steps for completing the square are as follows:
1. Move the constant term to the other side of the equation:
5x^2 + 18x = -11
2. Divide the coefficient of x^2 by 2 and square it:
(18/2)^2 = 81
3. Add the result from step 2 to both sides of the equation:
5x^2 + 18x + 81 = -11 + 81
5x^2 + 18x + 81 = 70
4. Factor the left side of the equation:
(√5x + 9)^2 = 70
5. Take the square root of both sides of the equation:
√[(√5x + 9)^2] = ±√70
6. Solve for x:
√5x + 9 = ±√70
√5x = -9 ±√70
5x = (-9 ±√70)^2
x = (-9 ±√70) / 5
Therefore, the solutions to the equation 5x^2 + 18x + 11 = 0 without using the quadratic formula are:
x = (-9 + √70) / 5
x = (-9 - √70) / 5